Sift flour, baking powder and salt together. Set aside.
To a separate bowl or stand mixer cream together softened butter and sugar until fluffy.
Add eggs and vanilla and continue beating until well incorporated.
Slowly add the dry ingredients to the wet and beat on low until everything is incorporated.
Place dough on a large piece of parchment paper that's been dusted with flour. Flour the top of the dough and cover with a second large piece of parchment paper. Press the dough down and then roll the dough out using a rolling pin to 1/2" thick.
Transfer the dough while still sandwiched between parchment paper to a large sheet pan or cutting board, just something large and flat to keep the dough from splitting while you place in the refrigerator to chill for 2 hours. **You can also cut the cookies out first and chill the individual cookies in the refrigerator for 2 hours.
Preheat oven to 350 degrees.
Cut out your first round of cookies if you haven't already. Use a flexible offset spatula to remove the cookies from the dough. Place the rest of the dough back in the refrigerator. Place cookies an inch apart on a parchment lined sheet pan and bake in rounds on the middle rack for 9-10 minutes. Let cookies cool on the cookie sheet for 5 minutes before transferring to a wire rack to cool completely.
Re-roll out the extra dough and cut more cookies. Follow this process until the dough is all used up.
While the cookies cool make the icing.
To a stand mixer with a whisk attachment or to a bowl using a hand mixer, beat softened butter until fluffy. Slowly add half the powdered sugar. Add the vanilla and half of the milk. Continue beating, adding the rest of the powdered sugar and milk. Beat until it's well incorporated, scraping the bowl down as needed.
Ice and decorate the cookies once they've completely cooled. Store in an air tight container.